rrxtopn - Mathbox for Glauco Siliprandi

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Theorem rrxtopn 40501

Description: The topology of the generalized real Euclidean space. (Contributed by Glauco Siliprandi, 24-Dec-2020.)

**Hypothesis**
Ref	Expression
rrxtopn.1	⊢ (𝜑 → 𝐼 ∈ 𝑉)

**Assertion**
Ref	Expression
rrxtopn	⊢ (𝜑 → (TopOpen‘(ℝ^‘𝐼)) = (MetOpen‘(𝑓 ∈ (Base‘(ℝ^‘𝐼)), 𝑔 ∈ (Base‘(ℝ^‘𝐼)) ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)))))))

Distinct variable groups: 𝑓,𝐼,𝑔,𝑥 𝑓,𝑉,𝑔,𝑥 Allowed substitution hints: 𝜑(𝑥,𝑓,𝑔)

**Proof of Theorem rrxtopn**
Step	Hyp	Ref	Expression
1		rrxtopn.1	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
2		eqid 2622	. . . . . 6 ⊢ (ℝ^‘𝐼) = (ℝ^‘𝐼)
3	2	rrxval 23175	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (ℝ^‘𝐼) = (toℂHil‘(ℝ_fld freeLMod 𝐼)))
4	1, 3	syl 17	. . . 4 ⊢ (𝜑 → (ℝ^‘𝐼) = (toℂHil‘(ℝ_fld freeLMod 𝐼)))
5	4	fveq2d 6195	. . 3 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝐼)) = (TopOpen‘(toℂHil‘(ℝ_fld freeLMod 𝐼))))
6		ovex 6678	. . . . 5 ⊢ (ℝ_fld freeLMod 𝐼) ∈ V
7		eqid 2622	. . . . . 6 ⊢ (toℂHil‘(ℝ_fld freeLMod 𝐼)) = (toℂHil‘(ℝ_fld freeLMod 𝐼))
8		eqid 2622	. . . . . 6 ⊢ (dist‘(toℂHil‘(ℝ_fld freeLMod 𝐼))) = (dist‘(toℂHil‘(ℝ_fld freeLMod 𝐼)))
9		eqid 2622	. . . . . 6 ⊢ (TopOpen‘(toℂHil‘(ℝ_fld freeLMod 𝐼))) = (TopOpen‘(toℂHil‘(ℝ_fld freeLMod 𝐼)))
10	7, 8, 9	tchtopn 23025	. . . . 5 ⊢ ((ℝ_fld freeLMod 𝐼) ∈ V → (TopOpen‘(toℂHil‘(ℝ_fld freeLMod 𝐼))) = (MetOpen‘(dist‘(toℂHil‘(ℝ_fld freeLMod 𝐼)))))
11	6, 10	ax-mp 5	. . . 4 ⊢ (TopOpen‘(toℂHil‘(ℝ_fld freeLMod 𝐼))) = (MetOpen‘(dist‘(toℂHil‘(ℝ_fld freeLMod 𝐼))))
12	11	a1i 11	. . 3 ⊢ (𝜑 → (TopOpen‘(toℂHil‘(ℝ_fld freeLMod 𝐼))) = (MetOpen‘(dist‘(toℂHil‘(ℝ_fld freeLMod 𝐼)))))
13	4	eqcomd 2628	. . . . 5 ⊢ (𝜑 → (toℂHil‘(ℝ_fld freeLMod 𝐼)) = (ℝ^‘𝐼))
14	13	fveq2d 6195	. . . 4 ⊢ (𝜑 → (dist‘(toℂHil‘(ℝ_fld freeLMod 𝐼))) = (dist‘(ℝ^‘𝐼)))
15	14	fveq2d 6195	. . 3 ⊢ (𝜑 → (MetOpen‘(dist‘(toℂHil‘(ℝ_fld freeLMod 𝐼)))) = (MetOpen‘(dist‘(ℝ^‘𝐼))))
16	5, 12, 15	3eqtrd 2660	. 2 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝐼)) = (MetOpen‘(dist‘(ℝ^‘𝐼))))
17		eqid 2622	. . . . . 6 ⊢ (Base‘(ℝ^‘𝐼)) = (Base‘(ℝ^‘𝐼))
18	2, 17	rrxds 23181	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (𝑓 ∈ (Base‘(ℝ^‘𝐼)), 𝑔 ∈ (Base‘(ℝ^‘𝐼)) ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))) = (dist‘(ℝ^‘𝐼)))
19	1, 18	syl 17	. . . 4 ⊢ (𝜑 → (𝑓 ∈ (Base‘(ℝ^‘𝐼)), 𝑔 ∈ (Base‘(ℝ^‘𝐼)) ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))) = (dist‘(ℝ^‘𝐼)))
20	19	eqcomd 2628	. . 3 ⊢ (𝜑 → (dist‘(ℝ^‘𝐼)) = (𝑓 ∈ (Base‘(ℝ^‘𝐼)), 𝑔 ∈ (Base‘(ℝ^‘𝐼)) ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))))
21	20	fveq2d 6195	. 2 ⊢ (𝜑 → (MetOpen‘(dist‘(ℝ^‘𝐼))) = (MetOpen‘(𝑓 ∈ (Base‘(ℝ^‘𝐼)), 𝑔 ∈ (Base‘(ℝ^‘𝐼)) ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)))))))
22	16, 21	eqtrd 2656	1 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝐼)) = (MetOpen‘(𝑓 ∈ (Base‘(ℝ^‘𝐼)), 𝑔 ∈ (Base‘(ℝ^‘𝐼)) ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 − cmin 10266 2c2 11070 ↑cexp 12860 √csqrt 13973 Basecbs 15857 distcds 15950 TopOpenctopn 16082 Σ_g cgsu 16101 MetOpencmopn 19736 ℝ_fldcrefld 19950 freeLMod cfrlm 20090 toℂHilctch 22967 ℝ^crrx 23171

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-fz 12327 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-topgen 16104 df-prds 16108 df-pws 16110 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-ghm 17658 df-cmn 18195 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-rnghom 18715 df-drng 18749 df-field 18750 df-subrg 18778 df-staf 18845 df-srng 18846 df-lmod 18865 df-lss 18933 df-sra 19172 df-rgmod 19173 df-psmet 19738 df-xmet 19739 df-bl 19741 df-mopn 19742 df-cnfld 19747 df-refld 19951 df-dsmm 20076 df-frlm 20091 df-top 20699 df-topon 20716 df-bases 20750 df-nm 22387 df-tng 22389 df-tch 22969 df-rrx 23173

This theorem is referenced by: rrxtopnfi 40506

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